Decoding for generalized orthogonal design for space-time codes for wireless communication

ABSTRACT

The prior art teachings for encoding signals and transmitting them over a plurality of antennas are advanced by disclosing a method for encoding for any number of transmitting antennas. Also disclosed is a generalized approach for maximum likelihood decoding where a decision rule is formed for all of the transmitting antennas of a transmitter, and a decision is made in favor of the transmitted symbols the minimize the equationwherert&lt;j &gt;is the signal received at time interval t, at receiving antenna j,h*epsilont(i)j is the complex conjugate of the channel transfer function between the transmitter antenna that is transmitting symbol ci and receiving antenna j, anddeltat(i) is the sign of symbol ci in time interval t.

REFERENCE TO RELATED APPLICATION

This is a continuation of application Ser. No. 09/186,907, filed Nov. 6,1998 now U.S. Pat. No. 6,088,408.

This application claims the benefit of U.S. Provisional Application No.60/065,095, filed Nov. 11, 1997; and of U.S. Provisional Application No.60/076,613, filed Mar. 3, 1998.

BACKGROUND OF THE INVENTION

This invention relates to wireless communication and, more particularly,to techniques for effective wireless communication in the presence offading and other degradations.

The most effective technique for mitigating multipath fading in awireless radio channel is to cancel the effect of fading at thetransmitter by controlling the transmitter's power. That is, if thechannel conditions are known at the transmitter (on one side of thelink), then the transmitter can pre-distort the signal to overcome theeffect of the channel at the receiver (on the other side). However,there are two fundamental problems with this approach. The first problemis the transmitter's dynamic range. For the transmitter to overcome an xdB fade, it must increase its power by x dB which, in most cases, is notpractical because of radiation power limitations, and the size and costof amplifiers. The second problem is that the transmitter does not haveany knowledge of the channel as seen by the receiver (except for timedivision duplex systems, where the transmitter receives power from aknown other transmitter over the same channel). Therefore, if one wantsto control a transmitter based on channel characteristics, channelinformation has to be sent from the receiver to the transmitter, whichresults in throughput degradation and added complexity to both thetransmitter and the receiver.

Other effective techniques are time and frequency diversity. Using timeinterleaving together with coding can provide diversity improvement. Thesame holds for frequency hopping and spread spectrum. However, timeinterleaving results in unnecessarily large delays when the channel isslowly varying. Equivalently, frequency diversity techniques areineffective when the coherence bandwidth of the channel is large (smalldelay spread).

It is well known that in most scattering environments antenna diversityis the most practical and effective technique for reducing the effect ofmultipath fading. The classical approach to antenna diversity is to usemultiple antennas at the receiver and perform combining (or selection)to improve the quality of the received signal.

The major problem with using the receiver diversity approach in currentwireless communication systems, such as IS-136 and GSM, is the cost,size and power consumption constraints of the receivers. For obviousreasons, small size, weight and cost are paramount. The addition ofmultiple antennas and RF chains (or selection and switching circuits) inreceivers is presently not be feasible. As a result, diversitytechniques have often been applied only to improve the up-link (receiverto base) transmission quality with multiple antennas (and receivers) atthe base station. Since a base station often serves thousands ofreceivers, it is more economical to add equipment to base stationsrather than the receivers

Recently, some interesting approaches for transmitter diversity havebeen suggested. A delay diversity scheme was proposed by A. Wittneben in“Base Station Modulation Diversity for Digital SIMULCAST,” Proceeding ofthe 1991 IEEE Vehicular Technology Conference (VTC 41 st), PP. 848-853,May 1991, and in “A New Bandwidth Efficient Transmit Antenna ModulationDiversity Scheme For Linear Digital Modulation,” in Proceeding of the1993 IEEE International Conference on Communications (IICC '93), PP.1630-1634, May 1993. The proposal is for a base station to transmit asequence of symbols through one antenna, and the same sequence ofsymbols—but delayed—through another antenna.

U.S. Pat. 5,479,448, issued to Nambirajan Seshadri on Dec. 26, 1995,discloses a similar arrangement where a sequence of codes is transmittedthrough two antennas. The sequence of codes is routed through a cyclingswitch that directs each code to the various antennas, in succession.Since copies of the same symbol are transmitted through multipleantennas at different times, both space and time diversity are achieved.A maximum likelihood sequence estimator (MLSE) or a minimum mean squarederror (MMSE) equalizer is then used to resolve multipath distortion andprovide diversity gain. See also N. Seshadri, J. H. Winters, “TwoSignaling Schemes for Improving the Error Performance of FDDTransmission Systems Using Transmitter Antenna Diversity,” Proceeding ofthe 1993 IEEE Vehicular Technology Conference (VTC 43rd), pp. 508-511,May 1993; and J. H. Winters, “The Diversity Gain of Transmit Diversityin Wireless Systems with Rayleigh Fading,” Proceeding of the 1994ICC/SUPERCOMM, New Orleans, Vol. 2, PP. 1121-1125, May 1994.

Still another interesting approach is disclosed by Tarokh, Seshadri,Calderbank and Naguib in U.S. application, Ser. No. 08/847635, filedApr. 25, 1997 (based on a provisional application filed Nov. 7, 1996),where symbols are encoded according to the antennas through which theyare simultaneously transmitted, and are decoded using a maximumlikelihood decoder. More specifically, the process at the transmitterhandles the information in blocks of M1 bits, where M1 is a multiple ofM2, i.e., M1=k*M2. It converts each successive group of M2 bits intoinformation symbols (generating thereby k information symbols), encodeseach sequence of k information symbols into n channel codes (developingthereby a group of n channel codes for each sequence of k informationsymbols), and applies each code of a group of codes to a differentantenna.

Recently, a powerful approach is disclosed by Alamouti et al in U.S.patent application Ser. No. 09/074,224, filed May 5, 1998, and titled“Transmitter Diversity Technique for Wireless Communication”. Thisdisclosure revealed that an arrangement with two transmitter antennascan be realized that provides diversity with bandwidth efficiency, easydecoding at the receiver (merely linear processing), and performancethat is the same as the performance of maximum ratio combiningarrangements. In this arrangement the constellation has four symbols,and a frame has two time slots during which two bits arrive. Those bitare encoded so that in a first time slot symbol c₁ and c₂ are sent bythe first and second antennas, respectively, and in a second time slotsymbols −c₂ * and c₁ * are sent by the first and second antennas,respectively. Accordingly, this can be expressed by an equation of theform r=Hc+n, where r is a vector of signals received in the two timeslots, c is a vector of symbols c₁ and c₂, n is a vector of receivednoise signals in the two time slots, and H is an orthogonal matrix thatreflects the above-described constellation of symbols.

The good performance of this disclosed approach forms an impetus forfinding other systems, with a larger number of transmit antennas, thathas equally good performance.

SUMMARY

The prior art teachings for encoding signals and transmitting them overa plurality of antennas are advanced by disclosing a method for encodingfor any number of transmitting antennas. Also disclosed is a generalizedapproach for maximum likelihood decoding where a decision rule is formedfor all of the transmitting antennas of a transmitter, and a decision ismade in favor of the transmitted symbols the minimize the equation$c_{i} = {{\arg \quad {\min\limits_{c}{{R_{i} - c}}^{2}}} + {\left( {{- 1} + {\sum{h_{i,j}}^{2}}} \right){c}^{2}}}$

where$R_{i} = {\sum\limits_{t = 1}^{n}\quad {\sum\limits_{j = 1}^{m}\quad {r_{t}^{j}h_{{ɛ_{t}{(i)}}j}^{*}{\delta_{t}(i)}}}}$

r_(t) ^(J) is the signal received at time interval t, at receivingantenna j,

h^(*) _(ε) _(t) _((i)j) is the complex conjugate of the channel transferfunction between the transmitter antenna that is transmitting symbolc_(i) and receiving antenna j, and δ_(t)(i) is the sign of symbol c_(i)in time interval t.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a transmitter having n antennas and areceiver having j antenna, where the transmitter and the receiveroperate in accordance with the principles disclosed herein;

DETAILED DESCRIPTION

FIG. 1 presents a block diagram of an arrangement with a transmitterhaving n transmitter antenna an a receiver with j receiving antenna.When n=2, FIG. 1 degenerates to FIG. 1 of the aforementioned Ser. No.09/074,224 Alamouti et al application. In that application an appliedsequence of symbols c₁,c₂,c₃,c₄,c₅,c₆ at the input of transmitter 10results in the following sequence being sent by antennas 11 and 12.

Time: t t + T t + 2T t + 3T t + 4T t + 5T Antenna 11 c₀ −c₁* c₂ −c₃* c₄−c₅* . . . Antenna 12 c₁ c₀* c₃ c₂* c₅ c₄* . . .

The transmission can be expressed by way of the matrix $\begin{matrix}{\begin{bmatrix}c_{1} & c_{2} \\{- c_{2}^{*}} & c_{1}^{*}\end{bmatrix},} & (1)\end{matrix}$

where the columns represent antennas, and the rows represent time oftransmission. The corresponding received signal (ignoring the noise) is:

Time: t t + T t + 2T t + 3T Antenna h₁c₁ + −h₁c₂ ^(*) + h₂c₁ ^(*) h₁c₃ +h₂c₄ −h₁c₄ ^(*) + h₂c₃ ^(*) ... 11 h₂c₂

where h₁ is the channel coefficient from antenna 11 to antenna 21, andh₂ is the channel coefficient from antenna 12 to antenna 21, which canalso be in the form $\begin{matrix}{{\begin{bmatrix}r_{1} \\r_{2}^{*}\end{bmatrix} = {\begin{bmatrix}h_{1} & h_{2} \\h_{2}^{*} & {- h_{1}^{*}}\end{bmatrix}\quad\begin{bmatrix}c_{1} \\c_{2}\end{bmatrix}}},\quad {{{or}\quad r} = {{Hc}.}}} & (2)\end{matrix}$

Extending this to n antennas at the base station and m antennas in theremote units, the signal r_(t) ^(j) represents the signal received attime t by antenna j, and it is given by $\begin{matrix}{r_{t}^{j} = {{\sum\limits_{i = 1}^{n}\quad {h_{ij}c_{t}^{j}}} + n_{t}^{j}}} & (3)\end{matrix}$

where n_(t) ^(j) is the noise at time t at receiver antenna j, and it isassumed to be a independent, zero mean, complex, Gaussian randomvariable. The average energy of the symbols transmitted by each of the nantennas is 1/n.

Assuming a perfect knowledge of the channel coefficients, h_(ij), fromtransmit antenna i to receive antenna j, the receiver's decision metricis $\begin{matrix}{\sum\limits_{t = 1}^{l}\quad {\sum\limits_{j = 1}^{m}\quad {{r_{t}^{j} - {\sum\limits_{i = 1}^{n}\quad {h_{ij}c_{t}^{j}}}}}^{2}}} & (4)\end{matrix}$

Over all codewords c₁ ¹c₁ ² . . . c₁ ^(n)c₂ ¹c₂ ² . . . c₂ ^(n) . . . c₁¹c₁ ² . . . c₁ ^(n) and decides in favor of the codeword that minimizesthis sum.

For a constellation with real symbols, what is desired is a matrix ofsize n that is orthogonal, with intermediates ±c₁,±c₂, . . . ±c_(n). Theexistence problem for orthogonal designs is known in the mathematicsliterature as the Hurwitz-Radon problem, and was completely settled byRadon at the beginning of the 20^(th) century. What has been shown isthat an orthogonal design exists if and only if n=2, 4 or 8.

Indeed, such a matrix can be designed for the FIG. 1 system for n=2, 4or 8, by employing, for example, the matrices $\begin{matrix}{\begin{bmatrix}c_{1} & c_{2} \\{- c_{2}} & c_{1}\end{bmatrix},} & (5) \\{\begin{bmatrix}c_{1} & c_{2} & c_{3} & c_{4} \\{- c_{2}} & c_{1} & {- c_{4}} & c_{3} \\{- c_{3}} & c_{4} & c_{1} & {- c_{2}} \\{- c_{4}} & {- c_{3}} & c_{2} & c_{1}\end{bmatrix}\quad {or}} & (6) \\{\begin{bmatrix}c_{1} & c_{2} & c_{3} & c_{4} & c_{5} & c_{6} & c_{7} & c_{8} \\{- c_{2}} & c_{1} & c_{4} & {- c_{3}} & c_{6} & {- c_{5}} & {- c_{8}} & c_{7} \\{- c_{3}} & {- c_{4}} & c_{1} & c_{2} & c_{7} & c_{8} & {- c_{5}} & {- c_{6}} \\{- c_{4}} & c_{3} & {- c_{2}} & c_{1} & c_{8} & {- c_{7}} & c_{6} & {- c_{5}} \\{- c_{5}} & {- c_{6}} & {- c_{7}} & {- c_{8}} & c_{1} & c_{2} & c_{3} & c_{4} \\{- c_{6}} & c_{5} & {- c_{8}} & c_{7} & {- c_{2}} & c_{1} & {- c_{4}} & c_{3} \\{- c_{7}} & c_{8} & c_{5} & {- c_{6}} & {- c_{3}} & c_{4} & c_{1} & {- c_{2}} \\{- c_{8}} & {- c_{7}} & c_{6} & c_{5} & {- c_{4}} & {- c_{3}} & c_{2} & c_{1}\end{bmatrix}.} & (7)\end{matrix}$

What that means, for example, is that when a transmitter employs 8antennas, it accumulates a frame of 8 bits and, with the beginning ofthe next frame, in the first time interval, the 8 antennas transmit bitsc₁,c₂,c₃,c₄,c₅,c₆,c₇,c₈ (the first row of symbols). During the secondtime interval, the 8 antennas transmit bits −c₂,c₁,c₄,−c₃,c₆,−c₅,−c₈,c₇(the second row of symbols), etc.

A perusal of the above matrices reveals that the rows are merepermutations of the first row, with possible different signs. Thepermutations can be denoted by ε_(k) (p) such that ε_(k)(p)=q means thatin row k, the symbol c_(p) appears in column q. The different signs canbe expressed by letting the sign of c_(i) in the k-th row be denoted byδ_(k)(i).

It can be shown that minimizing the metric of equation (4) is equivalentto minimizing the following sum $\begin{matrix}{\sum\limits_{i = 1}^{n}\quad \left( {{{{\sum\limits_{t = 1}^{l}\quad {\sum\limits_{j = 1}^{m}\quad {r_{t}^{j}h_{{ɛ_{i}{(i)}},j}^{*}\quad {\delta_{t}(i)}}}} - c_{i}}}^{2} + {\left( {{- 1} + {\sum\quad {h_{i,j}}^{2}}} \right){c_{i}}^{2}}} \right)} & (8)\end{matrix}$

Since the term${{{\sum\limits_{t = 1}^{l}\quad {\sum\limits_{j = 1}^{m}\quad {r_{t}^{j}h_{{ɛ_{t}{(i)}},j}^{*}\quad {\delta_{t}(i)}}}} - c_{i}}} + {\left( {{- 1} + {\sum\quad {h_{i,j}}^{2}}} \right){c_{i}}^{2}}$

only depends on c_(i), on the channel coefficients, and on thepermutations and signs of the matrix, it follows that minimizing theouter sum (over the summing index i) amounts to minimizing each of theterms for 1≦i≦n. Thus, the maximum likelihood detection rule is to formthe decision variable $\begin{matrix}{R_{i} = {\sum\limits_{i = 1}^{n}\quad {\sum\limits_{j = 1}^{m}\quad {r_{t}^{j}h_{{ɛ_{t}{(i)}},j}^{*}\quad {\delta_{t}(i)}}}}} & (9)\end{matrix}$

for all transmitting antennas, i=1,2, . . . n, and decide in favor of ismade in favor of symbol c_(i) from among all constellation symbols if$\begin{matrix}{c_{i} = {{\arg \quad {\min\limits_{c}{{R_{i} - c}}^{2}}} + {\left( {{- 1} + {\sum{h_{i,j}}^{2}}} \right){{c}^{2}.}}}} & (10)\end{matrix}$

This is a very simple decoding strategy that provides diversity.

There are two attractions in providing transmit diversity via orthogonaldesigns.

There is no loss in bandwidth, in the sense that orthogonal designsprovide the maximum possible transmission rate at full diversity.

There is an extremely simple maximum likelihood decoding algorithm whichonly uses linear combining at the receiver. The simplicity of thealgorithm comes from the orthogonality of the columns of the orthogonaldesign.

The above properties are preserved even if linear processing at thetransmitter is allowed. Therefore, in accordance with the principlesdisclosed herein, the definition of orthogonal arrays is relaxed toallow linear processing at the transmitter. Signals transmitted fromdifferent antennas will now be linear combinations of constellationsymbols.

The following defines a Hurwitz-Radon family of matrices.

Defintion: A set of n×n real matrices {B₁,B₂, . . . B_(k)} is called asize k Hurwitz-Radon family of matrices if

B _(i) ^(T) B _(i) =I

B _(i) ^(T) =−B _(i) , i=1,2, . . . ,k

B _(i) B _(j) =−B _(j) B _(i), 1≦i<j≦k.  (11)

It has been shown by Radon that when n=2^(a)b, where b is odd and a=4c+dwith 0≦d <4 and 0<c, then and Hurwitz-Radon family of n×n matricescontains less than ρ(n)=8c+2^(d)≦n matrices (the maximum number ofmember in the family is ρ(n)−1). A Hurwitz-Radon family that containsn−1 matrices exists if and only if n=2,4, or 8.

Definition: Let A be a p×q matrix with terms a_(ij), and let B be anyarbitrary matrix. The tensor product A{circle around (x)}B is given by$\begin{matrix}{\begin{bmatrix}{a_{11}B} & {a_{12}B} & \cdots & {a_{1q}B} \\a_{21B} & {a_{22}B} & \cdots & {a_{2q}B} \\\vdots & \vdots & ⋰ & \vdots \\{a_{p1}B} & {a_{p2}B} & \cdots & {a_{pq}B}\end{bmatrix}.} & (12)\end{matrix}$

Lemma: For any n there exists a Hurwitz-Radon family of matrices of sizeρ(n)−1 whose members are integer matrices in the set {−1,0,1}.

Proof: The proof is by explicit construction. Let I_(b) denote theidentity matrix of size b. We first notice that if n=2_(a)b with b odd,then since ρ(n) is independent of b (ρ(n)=8c+2^(d)) it follows thatρ(n)=ρ(2^(a)). Moreover, given a family of 2^(a)×2^(a) Hurwitz-Radoninteger matrices {A₁,A₂, . . . A_(k)} of size s=ρ(2^(a))−1, the set{A₁{circle around (X)}I_(h), . . . A_(k){circle around (X)}I_(h)} is aHurwitz-Radon family of n×n integer matrices of size ρ(n)−1. In light ofthis observation, it suffices to prove the lemma for n=2^(a). To thisend, we may choose a set of Hurwitz-Radon matrices, such as$\begin{matrix}{{R = \begin{bmatrix}0 & 1 \\{- 1} & 0\end{bmatrix}},} & (13) \\{{P = \begin{bmatrix}0 & 1 \\1 & 0\end{bmatrix}},\quad {and}} & (14) \\{{Q = \begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix}},} & (15)\end{matrix}$

and let n₁=s^(4s+3), n₂=s^(4s+4), n₃=s^(4s+5), n₄=s^(4s+6), andn₅=s^(4s+7). Then,

ρ(n ₂)=ρ(n ₁)+1

ρ(n ₃)=ρ(n ₁)+2

ρ(n ₄)=ρ(n ₁)+4

ρ(n ₅)=ρ(n ₁)+8.  (16)

One can observe that matrix R is a Hurwitz-Radon integer family of sizeρ(2)−1, {R{circle around (X)}I₂,P{circle around (X)}I₂, . . . Q{circlearound (X)}I₂} is a Hurwitz-Radon integer family of size ρ(2²)−1, and{I₂{circle around (X)}R{circle around (X)}I₂,I₂{circle around(X)}P{circle around (X)}R,Q{circle around (X)}Q{circle around(X)}R,P{circle around (X)}Q{circle around (X)}R,R{circle around(X)}P{circle around (X)}Q,R{circle around (X)}P{circle around(X)}P,R{circle around (X)}Q{circle around (X)}I₂} is anintegerHurwitz-Radon family of size ρ(2³)−1. Extending from the above,one can easily verify that if {A₁,A_(2, . . . A) _(k)} is an integerHurwitz-Radon family of n×n matrices, then

{R{circle around (X)}I _(n) }∪{Q{circle around (X)}A _(i) ,i=1,2, . . ., s}  (17)

is an integer Hurwitz-Radon family of s+1 integer matrices (2n×2n).

If, in addition, {L₁, L₂, . . . L_(m)} is an integer Hurwitz-Radonfamily of k×k matrices, then

{P{circle around (X)}I _(k) {circle around (X)}A _(i) ,i=1,2, . . . ,s}∪{ Q{circle around (X)}L _(j) {circle around (X)}I _(n) ,j=1,2, . . ., j}∪{R{circle around (X)}I _(nk)}  (18)

is an integer Hurewitz-Radon family of s+m+1 integer matrices (2nk×2nk).

With a family of integer Hurwitz-Radon matrices with size ρ(2³⁾⁻1constructed for n=2³, with entries in the set {−1, 0, 1}, equation (17)gives the transition from n to n₂. By using (18) and letting k=n₁ andn=2, we get the transition from n₁ to n₃. Similarly, with k=n₁ and n=4we get the transition from n₁ to n₃, and with k=n₁ and n=8 we get thetransition from n₁ to n₅.

The simple maximum likelihood decoding algorithm described above isachieved because of the orthogonality of columns of the design matrix.Thus, a more generalized definition of orthogonal design may betolerated. Not only does this create new and simple transmission schemesfor any number of transmit antennas, but also generalizes theHurwitz-Radon theory to non-square matrices.

Definition: A generalized orthogonal design of size n is a p×n matrixwith entries 0,±x₁,±x₂, . . . ±x_(k) such that ^(T)= is a diagonalmatrix with diagonal _(ii),i=1,2, . . . n of the form (l₁ ^(i)x₁ ²+l₂^(i)x₂ ²+ . . . +l_(k) ^(i)x_(k) ²). The coefficients l₁ ^(i),l₂ ^(i), .. . , l_(k) ^(i), are positive integers. The rate of is R=k/p.

Theorem: A p×n generalized orthogonal design in variables x₁, ,x₂, . . .x_(k) exists if and only if there exists a generalized orthogonal designin the same variables and of the same size such that ^(T)=(x₁ ²+x₁ ²+ .. . x_(k) ²)I.

In view of the above theorem, without loss of generality, one can assumethat any p×n generalized orthogonal design in variable x₁, x₂, . . .x_(k) satisfies

^(T)=(x ₁ ² +x ₁ ² + . . . x _(k) ²)I

The above derivations can be employed for transmitting signals from nantennas using a generalized orthogonal design.

Considering a constellation of size 2^(b), a throughput of kb/p can beachieved. At time slot 1, kb bits arrive at the encoder, which selectsconstellation symbols c₁,c₂, . . . c_(n). The encoder populates thematrix by setting x_(i)=c_(i), and at times t=1,2, . . . , p the signals_(t1),_(t2), . . . _(tn) are transmitted simultaneously from antennas1,2, . . . ,n. That is the transmission matrix design is $\begin{matrix}{ = {\begin{bmatrix}_{11} & _{12} & \cdots & _{1n} \\_{21} & _{22} & \cdots & _{2n} \\\vdots & \vdots & ⋰ & \vdots \\_{p1} & _{p2} & \cdots & _{pn}\end{bmatrix}.}} & (19)\end{matrix}$

Thus, kb bits are sent during each frame of p transmissions. It can beshown that the diversity order is nm. The theory of space-time codingsays that for a diversity order of nm, it is possible to transmit b bitsper time slot, and this is the best possible. Therefore, the rate R isdefined for this coding scheme is kb/pb, or k/p.

The following presents an approach for constructing high rate linearprocessing designs with low decoding complexity and full diversityorder. It is deemed advantageous to take transmitter memory intoaccount, and that means that given the rate, R, and the number oftransmitting antennas, n, it is advantageous to minimize the number oftime slots in a frame, p.

Definition: For a given pair (R,n), A(R,n) is the minimum number p suchthat there exists a p×n generalized design with rate at least. If nosuch design exists, then A(R,n)=∞.

The value of A(R,n) is the fundamental question of generalized designtheory. The most interesting part of this question is the computation ofA(1,n) since the generalized designes of full rate are bandwidthefficient. To address the question the following construction isoffered.

Construction I: Let X=(x₁,x₂, . . . ,x_(p)) and n≅ρ(p). In thediscussion above, a family of integer p×p matrices with ρ(p)−1 withmembers {A₁,A₂, . . . ,A_(ρ(p)−1)} was constructed (Lemma followingequation 12). That is, the members A_(i) are in the set {−1,0,1}. LetA₀=I and consider the p×n matrix whose j-th column is A_(j−1)X^(T) forj=1,2, . . . ,n. The Hurwitz-Radon conditions imply that is ageneralized orthogonal design of full rate.

From the above, a number of facts can be ascertained:

The value A(1,n) is the smaller number p such that n<ρ(p).

The value of A(1,n) is a power of 2 for any n≧2.

The value A(1,n)=min(2^(4c+d)) where the minimization is taken over theset {c,d|0≦c,0≦d<4 and 8c+2^(d)≧n}.

A(1,2)=2,A(1,3)=A(1,4)=4, and A(1,n)=8 for 5≦n≦8.

Orthogonal designs are delay optical for n=2,4, and 8.

For any R, A(R,n)<∞.

The above explicitly constructs a Hurwitz-Radon family of matrices ofsize p with ρ(p) members such that all the matrices in the family haveentries in the set {−1,0,1}. Having such a family of Hurwitz-Radonmatrices of size p=A(1,n), we can apply Construction I to provide a p×ngeneralized orthogonal design with full rate.

This full rate generalized orthogonal design has entries of the form±c₁,±c₂, . . . ,±c_(p). Thus, for a transmitter having n<8 transmitantennas the following optimal generalized designs of rate one are:$\begin{matrix}{{_{3} = \begin{bmatrix}c_{1} & c_{2} & c_{3} \\{- c_{2}} & c_{1} & {- c_{4}} \\{- c_{3}} & c_{4} & c_{1} \\{- c_{4}} & {- c_{3}} & c_{2}\end{bmatrix}},} & (21) \\{{_{5} = \begin{bmatrix}c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \\{- c_{2}} & c_{1} & c_{4} & {- c_{3}} & c_{6} \\{- c_{3}} & {- c_{4}} & c_{1} & c_{2} & c_{7} \\{- c_{4}} & c_{3} & {- c_{2}} & c_{1} & c_{8} \\{- c_{5}} & {- c_{6}} & {- c_{7}} & {- c_{8}} & c_{1} \\{- c_{6}} & c_{5} & {- c_{8}} & c_{7} & {- c_{2}} \\{- c_{7}} & c_{8} & c_{5} & {- c_{6}} & {- c_{3}} \\{- c_{8}} & {- c_{7}} & c_{6} & c_{5} & {- c_{4}}\end{bmatrix}},} & (22) \\{{_{6} = \begin{bmatrix}c_{1} & c_{2} & c_{3} & c_{4} & c_{5} & c_{6} \\{- c_{2}} & c_{1} & c_{4} & {- c_{3}} & c_{6} & {- c_{5}} \\{- c_{3}} & {- c_{4}} & c_{1} & c_{2} & c_{7} & c_{8} \\{- c_{4}} & c_{3} & {- c_{2}} & c_{1} & c_{8} & {- c_{7}} \\{- c_{5}} & {- c_{6}} & {- c_{7}} & {- c_{8}} & c_{1} & c_{2} \\{- c_{6}} & c_{5} & {- c_{8}} & c_{7} & {- c_{2}} & c_{1} \\{- c_{7}} & c_{8} & c_{5} & {- c_{6}} & {- c_{3}} & c_{4} \\{- c_{8}} & {- c_{7}} & c_{6} & c_{5} & {- c_{4}} & {- c_{3}}\end{bmatrix}},\quad {and}} & (23) \\{_{7} = {\begin{bmatrix}c_{1} & c_{2} & c_{3} & c_{4} & c_{5} & c_{6} & c_{7} \\{- c_{2}} & c_{1} & c_{4} & {- c_{3}} & c_{6} & {- c_{5}} & {- c_{8}} \\{- c_{3}} & {- c_{4}} & c_{1} & c_{2} & c_{7} & c_{8} & {- c_{5}} \\{- c_{4}} & c_{3} & {- c_{2}} & c_{1} & c_{8} & {- c_{7}} & c_{6} \\{- c_{5}} & {- c_{6}} & {- c_{7}} & {- c_{8}} & c_{1} & c_{2} & c_{3} \\{- c_{6}} & c_{5} & {- c_{8}} & c_{7} & {- c_{2}} & c_{1} & {- c_{4}} \\{- c_{7}} & c_{8} & c_{5} & {- c_{6}} & {- c_{3}} & c_{4} & c_{1} \\{- c_{8}} & {- c_{7}} & c_{6} & c_{5} & {- c_{4}} & {- c_{3}} & c_{2}\end{bmatrix}.}} & (24)\end{matrix}$

The simple transmit diversity schemes disclosed above are for a realsignal constellation. A design for a complex constellation is alsopossible. A complex orthogonal design of size n that is contemplatedhere is a unitary matrix whose entries are indeterminates ±c₁,±c₂, . . ., ±c_(n), their complex conjugates ±c₁*,±c₂*, . . . , ±c_(n)*, or theseindeterminates multiplied by ±i, where i={square root over (−1)}.Without loss of generality, we may select the first row to be c₁,c₂, . .. ,c_(n).

It can be shown that half rate (R=0.5) complex generalized orthogonaldesigns exist. They can be constructed by creating a design as describedabove for real symbols, and repeat the rows, except that each symbol isreplaced by its complex conjugate. Stated more formally, given that adesign needs to be realized for complex symbols, we can replace eachcomplex variable c_(i)=c_(i)+ic_(i), where i={square root over (−1)} bythe 2×2 real matrix $\begin{bmatrix}c_{i}^{} & c_{i}^{\mathcal{I}} \\{- c_{i}^{\mathcal{I}}} & c_{i}^{}\end{bmatrix}.$

In this way, $c_{i}^{*} = {{\begin{bmatrix}c_{i}^{} & {- c_{i}^{\mathcal{I}}} \\c_{i}^{\mathcal{I}} & c_{i}^{}\end{bmatrix}\quad {and}\quad {ic}_{i}} = {\begin{bmatrix}{- c_{i}^{}} & c_{i}^{\mathcal{I}} \\{- c_{i}^{\mathcal{I}}} & {- c_{i}^{}}\end{bmatrix}.}}$

It is easy to see that a matrix formed in this way is a real orthogonaldesign. The following presents half rate codes for transmission usingthree and four transmit antennas by, of course, an extension to anynumber of transmitting antennas follows directly from application of theprinciples disclosed above. $\begin{matrix}{{_{c}^{3} = \begin{bmatrix}c_{1} & c_{2} & c_{3} \\{- c_{2}} & c_{1} & {- c_{4}} \\{- c_{3}} & c_{4} & c_{1} \\{- c_{4}} & {- c_{3}} & c_{2} \\c_{1}^{*} & c_{2}^{*} & c_{3}^{*} \\{- c_{2}^{*}} & c_{1}^{*} & {- c_{4}^{*}} \\{- c_{3}^{*}} & c_{4}^{*} & c_{1}^{*} \\{- c_{4}^{*}} & {- c_{3}^{*}} & c_{2}^{*}\end{bmatrix}},} & (25) \\{_{c}^{4} = {\begin{bmatrix}\begin{matrix}c_{1} & c_{2} & c_{3} & c_{4} \\{- c_{2}} & c_{1} & c_{4} & {- c_{3}} \\{- c_{3}} & {- c_{4}} & c_{1} & c_{2} \\{- c_{4}} & c_{3} & {- c_{2}} & c_{1}\end{matrix} \\\begin{matrix}c_{1}^{*} & c_{2}^{*} & c_{3}^{*} & c_{4}^{*} \\{- c_{2}^{*}} & c_{1}^{*} & c_{4}^{*} & {- c_{3}^{*}} \\{- c_{3}^{*}} & {- c_{4}^{*}} & c_{1}^{*} & c_{2}^{*} \\{- c_{4}^{*}} & c_{3}^{*} & {- c_{2}^{*}} & c_{1}^{*}\end{matrix}\end{bmatrix}.}} & (26)\end{matrix}$

These transmission schemes and their analogs for higher values of n notonly give full diversity but give 3 dB extra coding gain over theuncoded, but they lose half of the theoretical bandwidth efficiency.

Some designs are available that provide a rate that is higher than 0.5.The following presents designs for rate 0.75 for n=3 and n=4.$\begin{matrix}{\begin{bmatrix}c_{1} & c_{2} & \frac{c_{3}}{\sqrt{2}} \\{- c_{2}^{*}} & c_{1}^{*} & \frac{c_{3}}{\sqrt{2}} \\\frac{c_{3}^{*}}{\sqrt{2}} & \frac{c_{3}^{*}}{\sqrt{2}} & \frac{\left( {{- c_{1}} - c_{1}^{*} + c_{2} - c_{2}^{*}} \right)}{2} \\\frac{c_{3}^{*}}{\sqrt{2}} & \frac{- c_{3}^{*}}{\sqrt{2}} & \frac{\left( {c_{2} + c_{2}^{*} + c_{1} - c_{1}^{*}} \right)}{2}\end{bmatrix}\quad {and}} & (27) \\{\begin{bmatrix}c_{1} & c_{2} & \frac{c_{3}}{\sqrt{2}} & \frac{c_{3}}{\sqrt{2}} \\{- c_{2}^{*}} & c_{1}^{*} & \frac{c_{3}}{\sqrt{2}} & \frac{- c_{3}}{\sqrt{2}} \\\frac{c_{3}^{*}}{\sqrt{2}} & \frac{c_{3}^{*}}{\sqrt{2}} & \frac{\left( {{- c_{1}} - c_{1}^{*} + c_{2} - c_{2}^{*}} \right)}{2} & \frac{\left( {{- c_{2}} - c_{2}^{*} + c_{1} - c_{1}^{*}} \right)}{2} \\\frac{c_{3}^{*}}{\sqrt{2}} & \frac{- c_{3}^{*}}{\sqrt{2}} & \frac{\left( {c_{2} + c_{2}^{*} + c_{1} - c_{1}^{*}} \right)}{2} & \frac{\left( {c_{1} + c_{1}^{*} + c_{2} - c_{2}^{*}} \right)}{2}\end{bmatrix}.} & (28)\end{matrix}$

FIG. 1 depicts an arrangement where a transmitter includes an encoder 13that is responsive to an applied steam of symbols. The encoder, in mostembodiments will include a memory for storing the incoming symbols.Those are processes in accordance with the above disclosure and,illustratively, are applied to n mappers 14. The mappers map the symbolsonto a two dimensional constellation, for example, and apply the mappedsymbols to n pulse shapers 15 which modulate the signals and apply themto transmitting antennas 11. The structure of transmitter 10 isillustrative only, and many other designs can be employed that wouldstill realize the benefits of this invention.

The transmitted signals are received by receiver 20, which includes jreceiving antennas 21. The received signals are applied to detector 25,which detect signals in accordance with, for example, the detectionscheme described above in connection with equations 9 and 10. Channelestimators 22 are conventional. Their function is to estimate thechannel parameters for detector 25.

We claim:
 1. A transmitter comprising: a module for developing acollection of n codes c1, c2, . . . cn, where n is an integer greaterthan 1, and for developing therefrom a matrix of codes, where one row ofsaid matrix contains said block of codes and other rows of said matrixcontain permutations of said codes, with at least one of said codesbeing multiplied by −1; a mapper module responsive to codes ofsuccessive rows of said matrix of codes, and n transmission units,responsive to said mapper module, each of which outputs a signalcorresponding to codes of a given column of said matrix of codes.
 2. Atransmitter according to claim 1, where said matrix of codes is aHurwitz-Radon matrix of order 2, 4, or
 8. 3. A transmitter according toclaim 1 where said matrix of codes results from a linear combination ofHurwitz-Radon matrices.
 4. A transmitter according to claim 1 where saidmatrix of codes is of the form $\begin{bmatrix}c_{1} & c_{2} \\{- c_{2}} & c_{1}\end{bmatrix}.$


5. A transmitter according to claim 1 where said matrix of codes is ofthe form $\begin{bmatrix}c_{1} & c_{2} & c_{3} & c_{4} \\{- c_{2}} & c_{1} & {- c_{4}} & c_{3} \\{- c_{3}} & c_{4} & c_{1} & {- c_{2}} \\{- c_{4}} & {- c_{3}} & c_{2} & c_{1}\end{bmatrix}.$


6. A transmitter according to claim 1 where said matrix of codes is ofthe form $\begin{bmatrix}c_{1} & c_{2} & c_{3} & c_{4} & c_{5} & c_{6} & c_{7} & c_{8} \\{- c_{2}} & c_{1} & c_{4} & {- c_{3}} & c_{6} & {- c_{5}} & {- c_{8}} & c_{7} \\{- c_{3}} & {- c_{4}} & c_{1} & c_{2} & c_{7} & c_{8} & {- c_{5}} & {- c_{6}} \\{- c_{4}} & c_{3} & {- c_{2}} & c_{1} & c_{8} & {- c_{1}} & c_{6} & {- c_{5}} \\{- c_{5}} & {- c_{6}} & {- c_{7}} & {- c_{8}} & c_{1} & c_{2} & c_{3} & c_{4} \\{- c_{6}} & c_{5} & {- c_{8}} & c_{7} & {- c_{2}} & c_{1} & {- c_{4}} & c_{3} \\{- c_{7}} & c_{8} & c_{5} & {- c_{6}} & {- c_{3}} & c_{4} & c_{1} & {- c_{2}} \\{- c_{8}} & {- c_{7}} & c_{6} & c_{5} & {- c_{4}} & {- c_{3}} & c_{2} & c_{1}\end{bmatrix}.$


7. A transmitter according to claim 1 were said matrix of codes is ofthe form $\begin{bmatrix}c_{1} & c_{2} & c_{3} \\{- c_{2}} & c_{1} & {- c_{4}} \\{- c_{3}} & c_{4} & c_{1} \\{- c_{4}} & {- c_{3}} & c_{2}\end{bmatrix}.$


8. A transmitter according to claim 1 where said matrix of codes is theform $\begin{bmatrix}c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \\{- c_{2}} & c_{1} & c_{4} & {- c_{3}} & c_{6} \\{- c_{3}} & {- c_{4}} & c_{1} & c_{2} & c_{7} \\{- c_{4}} & c_{3} & {- c_{2}} & c_{1} & c_{8} \\{- c_{5}} & {- c_{6}} & {- c_{7}} & {- c_{8}} & c_{1} \\{- c_{6}} & c_{5} & {- c_{8}} & c_{7} & {- c_{2}} \\{- c_{7}} & c_{8} & c_{5} & {- c_{6}} & {- c_{3}} \\{- c_{8}} & {- c_{7}} & c_{6} & c_{5} & {- c_{4}}\end{bmatrix}.$


9. A transmitter according to claim 1 where said matrix of codes is theform $\begin{bmatrix}c_{1} & c_{2} & c_{3} & c_{4} & c_{5} & c_{6} \\{- c_{2}} & c_{1} & c_{4} & {- c_{3}} & c_{6} & {- c_{5}} \\{- c_{3}} & {- c_{4}} & c_{1} & c_{2} & c_{7} & c_{8} \\{- c_{4}} & c_{3} & {- c_{2}} & c_{1} & c_{8} & {- c_{7}} \\{- c_{5}} & {- c_{6}} & {- c_{7}} & {- c_{8}} & c_{1} & c_{2} \\{- c_{6}} & c_{5} & {- c_{8}} & c_{7} & {- c_{2}} & c_{1} \\{- c_{7}} & c_{8} & c_{5} & {- c_{6}} & {- c_{3}} & c_{4} \\{- c_{8}} & {- c_{7}} & c_{6} & c_{5} & {- c_{4}} & {- c_{3}}\end{bmatrix}.$


10. A transmitter according to claim 1 where said matrix of codes is theform $\begin{bmatrix}c_{1} & c_{2} & c_{3} & c_{4} & c_{5} & c_{6} & c_{7} \\{- c_{2}} & c_{1} & c_{4} & {- c_{3}} & c_{6} & {- c_{5}} & {- c_{8}} \\{- c_{3}} & {- c_{4}} & c_{1} & c_{2} & c_{7} & c_{8} & {- c_{5}} \\{- c_{4}} & c_{3} & {- c_{2}} & c_{1} & c_{8} & {- c_{7}} & c_{6} \\{- c_{5}} & {- c_{6}} & {- c_{7}} & {- c_{8}} & c_{1} & c_{2} & c_{3} \\{- c_{6}} & c_{5} & {- c_{8}} & c_{7} & {- c_{2}} & c_{1} & {- c_{4}} \\{- c_{7}} & c_{8} & c_{5} & {- c_{6}} & {- c_{3}} & c_{4} & c_{1} \\{- c_{8}} & {- c_{7}} & c_{6} & c_{5} & {- c_{4}} & {- c_{3}} & c_{2}\end{bmatrix}.$


11. A transmitter according to claim 1 where said matrix of codes is theform $\begin{bmatrix}c_{1} & c_{2} & c_{3} \\{- c_{2}} & c_{1} & {- c_{4}} \\{- c_{3}} & c_{4} & c_{1} \\{- c_{4}} & {- c_{3}} & c_{2} \\c_{1}^{*} & c_{2}^{*} & c_{3}^{*} \\{- c_{2}^{*}} & c_{1}^{*} & {- c_{4}^{*}} \\{- c_{3}^{*}} & c_{4}^{*} & c_{1}^{*} \\{- c_{4}^{*}} & {- c_{3}^{*}} & c_{2}^{*}\end{bmatrix}.$


12. A transmitter according to claim 1 where said matrix of codes is theform $\begin{bmatrix}c_{1} & c_{2} & c_{3} & c_{4} \\{- c_{2}} & c_{1} & {- c_{4}} & c_{3} \\{- c_{3}} & c_{4} & c_{1} & {- c_{2}} \\{- c_{4}} & {- c_{3}} & c_{2} & c_{1} \\c_{1}^{*} & c_{2}^{*} & c_{3}^{*} & c_{4}^{*} \\{- c_{2}^{*}} & c_{1}^{*} & {- c_{4}^{*}} & c_{3}^{*} \\{- c_{3}^{*}} & c_{4}^{*} & c_{1}^{*} & {- c_{2}^{*}} \\{- c_{4}^{*}} & {- c_{3}^{*}} & c_{2}^{*} & c_{1}^{*}\end{bmatrix}.$


13. A transmitter according to claim 1 where said matrix of codes is theform $\begin{bmatrix}c_{1} & c_{2} & \frac{c_{3}}{\sqrt{2}} \\{- c_{2}^{*}} & c_{1}^{*} & \frac{c_{3}}{\sqrt{2}} \\\frac{c_{3}^{*}}{\sqrt{2}} & \frac{c_{3}^{*}}{\sqrt{2}} & \frac{\left( {{- c_{1}} - c_{1}^{*} + c_{2} - c_{2}^{*}} \right)}{2} \\\frac{c_{3}^{*}}{\sqrt{2}} & \frac{- c_{3}^{*}}{\sqrt{2}} & \frac{\left( {c_{2} + c_{2}^{*} + c_{1} - c_{1}^{*}} \right)}{2}\end{bmatrix}.$


14. A transmitter according to claim 1 where said matrix of codes is theform $\begin{bmatrix}c_{1} & c_{2} & \frac{c_{3}}{\sqrt{2}} & \frac{c_{3}}{\sqrt{2}} \\{- c_{2}^{*}} & c_{1}^{*} & \frac{c_{3}}{\sqrt{2}} & \frac{- c_{3}}{\sqrt{2}} \\\frac{c_{3}^{*}}{\sqrt{2}} & \frac{c_{3}^{*}}{\sqrt{2}} & \frac{\left( {{- c_{1}} - c_{1}^{*} + c_{2} - c_{2}^{*}} \right)}{2} & \frac{\left( {{- c_{2}} - c_{2}^{*} + c_{1} - c_{1}^{*}} \right)}{2} \\\frac{c_{3}^{*}}{\sqrt{2}} & \frac{- c_{3}^{*}}{\sqrt{2}} & \frac{\left( {c_{2} + c_{2}^{*} + c_{1} - c_{1}^{*}} \right)}{2} & \frac{\left( {c_{1} + c_{1}^{*} + c_{2} - c_{2}^{*}} \right)}{2}\end{bmatrix}.$


15. A transmitter comprising: a module for developing a block of symbolsc₁,c₂, . . . c_(n), and for developing therefrom a matrix of codesb_(ij), where a) subscript i designates the row of code b_(ij) in saidmatrix and subscript j designates the column of code b_(ij) in saidmatrix, b) said matrix has at least two columns, c) one column of saidmatrix contains said block of symbols with at least one of the codescorresponding to one of said symbols multiplied by −1, d) other columnsof said matrix contain permutations of said codes multiplied by p, wherep is selected to be +1 or −1 to yield$0 = {\sum\limits_{i = 0}^{N}\quad {b_{ij}b_{ik}}}$

 where j≠k, and a plurality of transmitting units, equal in number tonumber of said columns of said matrix, each of which outputs a signalcorresponding to codes of a given column in said matrix of codes.
 16. Atransmitter according to claim 15 where said matrix of codes is of theform $\begin{bmatrix}c_{1} & c_{2} & c_{3} & c_{4} \\{- c_{2}} & c_{1} & {- c_{4}} & c_{3} \\{- c_{3}} & c_{4} & c_{1} & {- c_{2}} \\{- c_{4}} & {- c_{3}} & c_{2} & c_{1}\end{bmatrix}.$


17. A transmitter according to claim 15 where said matrix of codes is ofthe form $\begin{bmatrix}c_{1} & {- c_{2}} & {- c_{3}} & {- c_{4}} \\c_{2} & c_{1} & c_{4} & {- c_{3}} \\c_{3} & {- c_{4}} & c_{1} & c_{2} \\c_{4} & c_{3} & {- c_{2}} & c_{1}\end{bmatrix}.$


18. A transmitter according to claim 15 where said matrix of codes ofthe form $\begin{bmatrix}c_{1} & {- c_{3}} \\c_{2} & c_{4} \\c_{3} & c_{1} \\c_{4} & {- c_{2}}\end{bmatrix}.$


19. A transmitter comprising: a module for developing a block of symbolsc₁,c₂, . . . c_(n), and for developing therefrom a matrix of codesb_(ij), where a) subscript i designates the row of code b_(ij) in saidmatrix and subscript j designates the column of code b_(ij) in saidmatrix, b) said matrix has at least two rows, c) one row of said matrixcontains said block of symbols with at least one of the codescorresponding to one of said symbols multiplied by −1, d) other rows ofsaid matrix contain permutations of said codes multiplied by p, where pis selected to be +1 or −1 to yield$0 = {\sum\limits_{i = 0}^{N}\quad {b_{ij}b_{kj}}}$

 where i≠k, and a plurality of transmitting units, equal in number tonumber of columns of said matrix, each of which outputs a signalcorresponding to codes of a given column in said matrix of codes.
 20. Atransmitter according to claim 15 where said matrix of codes of the form$\begin{bmatrix}c_{1} & {- c_{2}} & {- c_{3}} & {- c_{4}} \\c_{2} & c_{1} & c_{4} & {- c_{3}} \\c_{3} & {- c_{4}} & c_{1} & c_{2} \\c_{4} & c_{3} & {- c_{2}} & c_{1}\end{bmatrix}.$


21. A transmitter according to claim 15 where said matrix of codes ofthe form $\begin{bmatrix}c_{1} & c_{2} & c_{3} & c_{4} \\{- c_{2}} & c_{1} & {- c_{4}} & c_{3} \\{- c_{3}} & c_{4} & c_{1} & {- c_{2}} \\{- c_{4}} & {- c_{3}} & c_{2} & c_{1}\end{bmatrix}.$


22. A transmitter according to claim 15 where said matrix of codesincludes the columns $\begin{bmatrix}c_{1} & c_{2} & c_{3} & c_{4} \\{- c_{3}} & c_{4} & c_{1} & {- c_{2}}\end{bmatrix}.$